| The thesis consists of three parts, all of which are concerned with approaches to the problem of filtering random processes or fields. However, methods employed differ substantially.; The first part builds on the well developed theory of linear Kalman filtering in order to adapt it to the setup and types of signals usually associated with non-parametric statistics. A class of signals possessing certain smoothness properties is considered, as well as cases with small noise and common noise. An adaptive filtering scheme is proposed which is then proved to be asymptotically equivalent to optimal kernel smoothing estimates usually used in non-parametric statistics.; The second part extends the traditional nonlinear filtering problem and the spectral separating scheme (S3) for an efficient numerical implementation in the case of distributed (field) observations. Connection of nonlinear filtering is exposed to matched filters commonly employed in engineering literature and practice.; Afterwards, the problem of increasing the precision of the spectral separating scheme is considered, and advantages of using wavelet bases in the scheme are discussed. Two examples arising from practical problems are presented.; The third part deals with modifications of the well-known Wald's (or Sequential Probability Ratio) Test and its applications to image processing. Reflected Wald's Test is introduced, consistency is proved for estimates based on it in the exponential and Gaussian noise cases. A new edge detection algorithm based on the Reflected Wald's Test is discussed. |
